Monotone convergence theorem

A monotone sequence, i.e. $a_n\ge a_{n+1}$ or $a_n\le a_{n+1}$ for all $n\in \mathbb{N}$, is convergent if and only if it is bounded. Furthermore,

• if $\{a_n\}$ is monotone increasing, $\{a_n\}\to \sup \{a_n\}$
• if $\{a_n\}$ is monotone decreasing, $\{a_n\}\to \inf \{a_n\}$

Proof